# Favourite problem books at university level

As background let me start by stating what I perceive to be the point of problem books, or to put the matter in perhaps more acceptable way, how I define problem books. A large majority of textbooks include exercises. Problem books have two distinctive features: the first is that they include at least hints and often complete solutions, ideally both; the second is that the problems are not routine exercises.

Exercises are primarily to make sure you have correctly grasped the notation, key results, key concepts etc. They are an essential test of understanding for the majority of students. On the other hand, a teacher or researcher in the field can typically solve exercises on sight. Certainly they can see how to solve them, even if some of the details might take a few minutes to work out. Most Questions on this site are exercises. The major group of exceptions (leaving aside the hopelessly ill-drafted questions) are the contest problems.

Note that the problem/exercise distinction is independent of level. Questions in several complex variable theory can be exercises, and questions about Euclidean geometry can be hard contest problems.

Contest problems have become an important subset of "problems" (under my usage of the term). However, a large majority of contest problems are at the teenage pre-university level (all the "olympiads" and similar contests). Two easily accessible exceptions are the Putnam exams and (less easily) the Miklos Schweitzer competitions.

Over the last decade or so, it has become more fashionable to publish problem books as supplementary material for university teaching. The earliest and best were perhaps the Polya/Szego Problems and Theorems in Analysis (2 vols) (still in print), and Halmos' A Hilbert Space Problem Book. As an undergraduate at Cambridge University in the late 1960s the Knopp problem books were assigned for the first complex variable course.

There are still some really excellent new(ish) problem books. Some of my favourites are Problem Book in Relativity and Gravitation (Lightman et al) - not so new (I see I bought my copy in 1977), Cosmology and Astrophysics through Problems (Padmanabhan) (purchased 1996), and Arnold's Problems (acquired 2005, but material originally published 1999).

Guy and Croft's books (Unsolved problems in ...) are not really the same. They are essentially convenient annotated references. But Chung and Graham's Erdos on Graphs is different somehow. I like it, despite my lack of progress with it.

I also like the multi-authored Algebraic Geometry, A problem solving approach, despite the fact that many problems are trivial. I found it a fast way into a field I had totally neglected. Similarly, Halmos' unique style carries me along in his more recent Linear Algebra Problem book despite the fact that many problems can be solved on sight.

Finally, I hesitate to mention names, but some of the problem books on number theory seem to me collections of exercises rather than problems.

So my questions are: (1) can anyone recommend any favourites to keep me amused on the dark winter evenings that are fast approaching ...? (2) clearly the viewpoint above is unashamedly elitist. I am worried about the future. Too many people are being brought up (right through toy problems at the PhD level) in a way that is unlikely to help them do worthwhile research. Is that viewpoint tenable? Should the best students (at least) be encouraged to tackle problems at the Putnam level as undergraduates? At the Miklos Schweitzer level? (3) is it feasible to give as homework, problems at the contest level? Or are they simply "enrichment material"? (4) The previous questions seem to be addressed primarily to teachers and researchers. What do students think? Would they like more problems? Or less.

Finally I am aware of several previous questions in this area. The closest seems to be Good problem books at a relatively advanced level? . But it is not the same question. Nor did it attract much interest :(

• At the risk of initiating an endless comment stream, I want to mention Radulescu's Problems in Real Analysis. Some of these involve mere ad hoc tricks and there is a definite competition focus, but many problems explore important subtleties of the fundamental concepts. Jul 3, 2016 at 20:54
• I think all the problems are problems, and the independent ones don't have solutions. But I don't have it in front of me. Jul 3, 2016 at 21:07
• The reference for very challenging real analysis problems at undergraduate and graduate levels is Makarov's Selected Problems in Real Analysis Jul 3, 2016 at 21:08

List of problem books that may fit the definition in the question:

(1) Polya/Szego (2 vols)

(2) Halmos: A Hilbert Space Problem Book

(3) Lightman et al: Problem Book in Relativity and Gravitation

(4) Padmanabhan: Cosmology and Astrophysics through problems

(5) Arnold's Problems

(6) Chung and Graham: Erdos on Graphs

(7) Miklos Schweitzer Contest (2 vols)

(8) Garrity et al: Algebraic Geometry, a problem solving approach

(9) Radulescou: Problems in Real Analysis [thanks to @symplectomorphic ]

(10) Makarov: Selected problems in Real Analysis [thanks to @LeGrandDODOM ]

(11) Knopp: Problem Book in the Theory of Functions (2 vols)

• @LeGrandDODOM Guess I will have to borrow (10) from the library, seems expensive ... Jul 4, 2016 at 9:29

This became too long for a comment, but it is only my viewpoint on your second and fourth question. I did include a fun problem to compensate for the long read.

I've been brought up with toy exercises as well. Even when I was 18 years old I never thought about anything to deeply. An answer either came quickly or I labeled it as too difficult and dismissed it altogether. When I was 19 years old, I first thought about how to find the roots of a quadratic equation since I had forgotten the formula. After 2 minutes or so, and writing stuff down, I 'discovered' the formula myself. I still forget the formula every now and then, but it takes ten seconds to derive the formula again.

Even though solving an actual problem feels good, I was used to toy exercises for the most part of my life. The first 3 years at university I was interested in mathematics but not used to solving anything beyond the routine exercises. I needed inspiring people and smart friends to help me on the right road and seriously think about problems. But unfortunately, 21 years of not having that mentality leaves deep scars. I am not the mathematician I could have been.

When you ask what students want, you probably are going to get two very different versions. Beginning students that have only seen toy exercises are unlikely to demand serious problems and work hard on them. You cannot really blame them since they know no better. The other answer is very likely to be a hindsight answer. In hindsight, thinking harder on certain problems and doing more work would have helped you becoming a better version of you.

Recently I started doing a Phd (with a large teaching assignment), I'm certainly not the most gifted mathematician but I do have my clear moments. Sometimes a smart insight occurs, but only after struggling with something and making my fair share of mistakes. This process is new to me and at times very hard. Having had only toy exercises myself for a long period of time certainly makes dealing with it during my phd for the first time a lot harder. This should not be the case!

More often than not do we hear old wise men say that things used to better. And, when it comes to educating mathematics, I have to agree. I see a lot of students struggling with exercises that even I did consider toy exercises myself when I was (a not so great) student myself. A vast majority did considerably less thinking than I did up to that point, and well, that's absolutely shocking. I try to motivate students and present them interesting problems that should help your overal understanding of mathematics. I try to tell the history of certain problems and the difficulties that great mathematicians had tryin to solve them. I hope that in this way I can save some students from not thinking, and not thinking is something that they were thought very well.

PS: How many numbers of the form $101$, $10101$, $1010101, \dots$ are prime?

• Well, they're all odd; that's a start:) And they're of the form $P_n=\sum_{i=0}^n4^n=1+\sum_{i=1}^n4^n$, which looks something like (but not a whole lot like) Mersenne primes. Maybe those $P_n$'s can be massaged into a form that's a closer match to something. (P.S. I thought the answer was 43 -- Hitchhiker's Guide?:) Jul 3, 2016 at 22:43
• @JohnForkosh: Those are not binary numbers, they are not of that form and the answer in 'A Hitchhiker's guide to the Galaxy' is $42$. Jul 3, 2016 at 22:53
• The number is $1+100+100^2+\cdots+100^n$. Now the usual formula for the sum of a geometric series almost finishes things, we get $\frac{100^{n+1}-1}{99}$, which is $(10^{n+1}-1)(10^{n+1}+1)/99$. Jul 4, 2016 at 0:27
• I never read Hitchhiker's. $42 = 101010_2$, so why isn't it 85? I guess I am too literal minded and spoil good jokes :( Jul 4, 2016 at 9:21
• On a less serious note, many thanks for the answer. Most interesting. Don't exactly know the protocol about accepting your own CW answer, so $\checkmark$. Jul 4, 2016 at 9:26